Question

Determine all positive integers $n$ for which $2^n+1$ is divisible by $3.$

Answer 1

$2^n+1$ can be written as $(3-1{)}^n+1$
$=[3^n+(-1{)}^n+amultipleof3]+1$
$=(-1{)}^n+1+amultipleof3$
Therefore $2^n+1$ is divisible by $3$ if and only if $(-1{)}^n+1$ is divisible by $3.$
If 'n' is even , $(-1{)}^n+1=2,$ which is not divisible by $3.$
If 'n' is odd , $(-1{)}^n+1=0,$ which is divisible by $3.$
Therefore $2^n+1$ is divisible by $3$ if and only if n is odd.